Related papers: Classical dynamical $r$-matrices for the Chern-Sim…
We study the conditions for classical r-matrices to be compatible with the generalised Chern-Simons action for 3d gravity. Compatibility means solving the classical Yang-Baxter equations with a prescribed symmetric part for each of the real…
We study the interplay between double cross sum decompositions of a given Lie algebra and classical r-matrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an…
We present a derivation of the dynamical r-matrices of the Calogero-Moser models using the Hamiltonian reduction procedure to get general formulae. We describe the dynamical r-matrices thus found for spin Calogero-Moser models and…
A dynamical $r$-matrix is associated with every self-dual Lie algebra $\A$ which is graded by finite-dimensional subspaces as $\A=\oplus_{n \in \cZ} \A_n$, where $\A_n$ is dual to $\A_{-n}$ with respect to the invariant scalar product on…
We provide a general study for triangular dynamical r-matrices using Poisson geometry. We show that a triangular dynamical r-matrix always gives rise to a regular Poisson manifold. Using the Fedosov method, we prove that non-degenerate…
A generalisation of the classical Calogero-Moser model obtained by coupling it to the Gaudin model is considered. The recently found classical dynamical r-matrix [E. Billey, J. Avan and O. Babelon, PAR LPTHE 93-55] for the…
We apply Dirac's gauge fixing procedure to Chern-Simons theory with gauge group ISO(2,1) on manifolds RxS, where S is a punctured oriented surface of general genus. For all gauge fixing conditions that satisfy certain structural…
We present further examples of the correspondence between solutions of type IIB supergravity and classical $r$-matrices satisfying the classical Yang-Baxter equation (CYBE). In the previous works, classical $r$-matrices have been composed…
The dynamical generalization of the classical Yang-Baxter equation that governs the possible Poisson structures on the space of chiral WZNW fields with generic monodromy is reviewed. It is explained that for particular choices of the chiral…
According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds…
In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present…
In this note we complement recent results on the exchange $r$-matrices appearing in the chiral WZNW model by providing a direct, purely finite-dimensional description of the relationship between the monodromy dependent 2-form that enters…
We briefly review the possible Poisson structures on the chiral WZNW phase space and discuss the associated Poisson-Lie groupoids. Many interesting dynamical r-matrices appear naturally in this framework. Particular attention is paid to the…
We consider a hierarchy of the natural type Hamiltonian systems of $n$ degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of $2\times 2$…
A simple cubic matrix model is presented, which has truncations that, it is argued, lead at the classical level to a variety of theories of gauge fields and gravity. These include Chern-Simons theory in d=3, and BF theory and general…
Using recent results of P. Etingof and A. Varchenko on the Classical Dynamical Yang-Baxter equation, we reduce the classification of dynamical r-matrices r on a commutative subalgebra l of a Lie algebra g to a purely algebraic problem under…
We use a Riemannian (or pseudo-Riemannian) geometric framework to formulate the theory of the classical r-matrix for integrable systems. In this picture the r-matrix is related to a fourth rank tensor, named the r-tensor, on the…
The algebraic structure underlying the classical $r$-matrix formulation of the complex sine-Gordon model is fully elucidated. It is characterized by two matrices $a$ and $s$, components of the $r$ matrix as $r=a-s$. They obey a modified…
We study the quantization of Chern-Simons theory with group $G$ coupled to dynamical sources. We first study the dynamics of Chern-Simons sources in the Hamiltonian framework. The gauge group of this system is reduced to the Cartan subgroup…
We suggest a formula for quantum universal $R$-matrices corresponding to quasitriangular classical $r$-matrices classified by Belavin and Drinfeld for all simple Lie algebras. The $R$-matrices are obtained by twisting the standard universal…